X and Y will be continuous random
variables which have joint probability density function

f

X;Y

( x, y) = c x y2 if 0 < y < x < 4 ; = 0 otherwise

where c is some non-negative constant. Do the following:
(a) Determine the constant c. Answer =
15/
45 :

(b) Calculate the marginal density function

fY of Y . Answer =
15
2 (45) [ 16 y2 -y4 ] :

(c) Calculate

P [ 4 < X + Y ] . Answer =
59/
64

:

(d) Calculate

P [ Y < 2 j 4 < X + Y ] . Answer =
12/
59

:

(e) Calculate

P [ 2 < X j Y = 1 ] . Answer =
4/
5

Hey guys, this is a question on one of my tutorial sheets, Ive worked through the first 4 sucessfully but cant seem to get the correct answer for (e). We were given answers in class annd told to workout ourselves. Im just not sure am I doing it correctly. Here are the steps I took.

1: I drew the region on a graph, the x- values ran from x = 2 to x = 4 (for the purpose of integration limits) and the y values ran from y = 1 to y = x.

2: I found the joint density which I got to be 5772/6(45)

3: I worked out the value of the marginal density of y at y=1. And then divided this number into the joint density to get the conditional density.

Im just wondering where I have gone wrong and if anyone could give a helping hand